Diagonal unitary entangling gates and contradiagonal quantum states

Nonlocal properties of an ensemble of diagonal random unitary matrices of order $N^2$ are investigated. The average Schmidt strength of such a bipartite diagonal quantum gate is shown to scale as $\log N$, in contrast to the $\log N^2$ behavior characteristic to random unitary gates. Entangling power of a diagonal gate $U$ is related to the von Neumann entropy of an auxiliary quantum state $\rho=AA^{\dagger}/N^2$, where the square matrix $A$ is obtained by reshaping the vector of diagonal elements of $U$ of length $N^2$ into a square matrix of order $N$. This fact provides a motivation to study the ensemble of non-hermitian unimodular matrices $A$, with all entries of the same modulus and random phases and the ensemble of quantum states $\rho$, such that all their diagonal entries are equal to $1/N$. Such a state is contradiagonal with respect to the computational basis, in sense that among all unitary equivalent states it maximizes the entropy copied to the environment due to the coarse graining process. The first four moments of the squared singular values of the unimodular ensemble are derived, based on which we conjecture a connection to a recently studied combinatorial object called the "Borel triangle". This allows us to find exactly the mean von Neumann entropy for random phase density matrices and the average entanglement for the corresponding ensemble of bipartite pure states.Comment: 14 pages, 6 figure

Maze solvers demystified and some other thoughts

There is a growing interest towards implementation of maze solving in spatially-extended physical, chemical and living systems. Several reports of prototypes attracted great publicity, e.g. maze solving with slime mould and epithelial cells, maze navigating droplets. We show that most prototypes utilise one of two phenomena: a shortest path in a maze is a path of the least resistance for fluid and current flow, and a shortest path is a path of the steepest gradient of chemoattractants. We discuss that substrates with so-called maze-solving capabilities simply trace flow currents or chemical diffusion gradients. We illustrate our thoughts with a model of flow and experiments with slime mould. The chapter ends with a discussion of experiments on maze solving with plant roots and leeches which show limitations of the chemical diffusion maze-solving approach.Comment: This is a preliminary version of the chapter to be published in Adamatzky A. (Ed.) Shortest path solvers. From software to wetware. Springer, 201

Parallel scheduling of recursively defined arrays

A new method of automatic generation of concurrent programs which constructs arrays defined by sets of recursive equations is described. It is assumed that the time of computation of an array element is a linear combination of its indices, and integer programming is used to seek a succession of hyperplanes along which array elements can be computed concurrently. The method can be used to schedule equations involving variable length dependency vectors and mutually recursive arrays. Portions of the work reported here have been implemented in the PS automatic program generation system

Absolute Entropy and Energy of Carbon Dioxide Using the Two-Phase Thermodynamic Model

The two-phase thermodynamic (2PT) model is used to determine the absolute entropy and energy of carbon dioxide over a wide range of conditions from molecular dynamics trajectories. The 2PT method determines the thermodynamic properties by applying the proper statistical mechanical partition function to the normal modes of a fluid. The vibrational density of state (DoS), obtained from the Fourier transform of the velocity autocorrelation function, converges quickly, allowing the free energy, entropy, and other thermodynamic properties to be determined from short 20-ps MD trajectories. The anharmonic effects in the vibrations are accounted for by the broadening of the normal modes into bands from sampling the velocities over the trajectory. The low frequency diffusive modes, which lead to finite DoS at zero frequency, are accounted for by considering the DoS as a superposition of gas-phase and solid-phase components (two phases). The analytical decomposition of the DoS allows for an evaluation of properties contributed by different types of molecular motions. We show that this 2PT analysis leads to accurate predictions of entropy and energy of CO_2 over a wide range of conditions (from the triple point to the critical point of both the vapor and the liquid phases along the saturation line). This allows the equation of state of CO_2 to be determined, which is limited only by the accuracy of the force field. We also validated that the 2PT entropy agrees with that determined from thermodynamic integration, but 2PT requires only a fraction of the time. A complication for CO_2 is that its equilibrium configuration is linear, which would have only two rotational modes, but during the dynamics it is never exactly linear, so that there is a third mode from rotational about the axis. In this work, we show how to treat such linear molecules in the 2PT framework

Convergence order of upwind type schemes for transport equations with discontinuous coefficients

An analysis of the error of the upwind scheme for transport equation with discontinuous coefficients is provided. We consider here a velocity field that is bounded and one-sided Lipschitz continuous. In this framework, solutions are defined in the sense of measures along the lines of Poupaud and Rascle's work. We study the convergence order of the upwind scheme in the Wasserstein distances. More precisely, we prove that in this setting the convergence order is 1/2. We also show the optimality of this result. In the appendix, we show that this result also applies to other "diffusive" "first order" schemes and to a forward semi-Lagrangian scheme

Simulating the weak death of the neutron in a femtoscale universe with near-Exascale computing

The fundamental particle theory called Quantum Chromodynamics (QCD) dictates everything about protons and neutrons, from their intrinsic properties to interactions that bind them into atomic nuclei. Quantities that cannot be fully resolved through experiment, such as the neutron lifetime (whose precise value is important for the existence of light-atomic elements that make the sun shine and life possible), may be understood through numerical solutions to QCD. We directly solve QCD using Lattice Gauge Theory and calculate nuclear observables such as neutron lifetime. We have developed an improved algorithm that exponentially decreases the time-to solution and applied it on the new CORAL supercomputers, Sierra and Summit. We use run-time autotuning to distribute GPU resources, achieving 20% performance at low node count. We also developed optimal application mapping through a job manager, which allows CPU and GPU jobs to be interleaved, yielding 15% of peak performance when deployed across large fractions of CORAL.Comment: 2018 Gordon Bell Finalist: 9 pages, 9 figures; v2: fixed 2 typos and appended acknowledgement

Англійська мова для студентів електромеханічних спеціальностей

Навчальний посібник розрахований на студентів напряму підготовки 6.050702 Електромеханіка. Містить уроки, що структуровані за тематичними розділами, граматичний коментар, короткі англо-український і українсько- англійський словники та додатки, які спрямовані на закріплення загальних навичок володіння англійською мовою. Акцентований на ɨсобливості термінології, що застосовується у науково-технічній галузі, зокрема, в електромеханіці та виконання запропонованих завдань, що буде сприяти формуванню навичок перекладу з англійської та української мов, сприйняттю письмової та усної англійської мови, вмінню письмового викладення англійською мовою науково-технічних та інших текстів під час професійної діяльності, спілкуванню з професійних та загальних питань тощо